One approach to obtain such an order is to describe it in FO logic—that is, to give a FO transduction which produces an ordering on any given tournament.
Indeed, FO transductions preserve twin-width up to some function [1]}.
Thus, if there is a universal transduction \(\Phi \)  which on any tournament \(T\)  computes some order \(<\) ,
we immediately obtain \(\operatorname{tww}(T,<) \leqslant f(\operatorname{tww}(T))\)  as desired.
With such a transduction, and some additional requirements such as \(<\)  being efficiently computable,
it would be straightforward to obtain our results from the known case of ordered graphs [2]}.
Unfortunately, this approach fails, as any formula transducing a total order from the
iterated blow-up of the 3-cycle needs to have size increasing in the number of iterations [3]}.
This class of counter-examples also has twin-width 1,
meaning that it is not even possible to transduce an order
on the class of tournaments of twin-width \(k\)  for a fixed \(k\) .
